Download Non-Standard Normal Distribution

QM 360-255-LW Exercises
Quantitative Methods II
Non-Standard Normal Distribution
Filename: E10) Non-Standard Normal Distribution.Doc
Carl F. Gauss
1.
Find the following areas under a normal distribution curve
with µ = 20 and s = 4.
a)
Area between x = 20 and x = 27
b) Area between x = 23 and x = 25
c)
Area between x = 9.5 and x = 17
2.
Determine the area of the normal distribution curve with
µ = 55 and s = 7.
a)
to the right of x = 58
b) to the right of x = 43
c)
to the left of x = 67
d) to the left of x = 24
3.
If a random variable has the normal distribution with
µ = 80.0 and s = 4.8, find the probabilities that it will take
on a value
a) less than 87.2;
b) greater than 76.4;
c) between 81.2 and 86.0;
d) between 71.6 and 88.4.
4.
If a random variable has the normal distribution with
µ = 62.5 and s = 12.4, find the probabilities that it will take
on a value
a) less than 40.1;
b) greater than 69.3;
c) between 65.0 and 75.0;
d) between 57.4 and 67.6.
5.
If a random variables has the normal distribution with µ = 102.4 and σ = 3.6, find the probabilities that it will take on a value
a) less than 107.6;
b) greater than 99.7;
c) between 106.9 and 110.5;
d) between 96.1 and 104.2.
6.
If the assembly time of an “easy to assemble” toy is a random variable having the normal distribution with µ = 12.8 minutes and
σ = 4.0 minutes, what are the probabilities that this kind of toy can be assembled in
a) less than 10.0 minutes;
b) anywhere from 11.0 to 14.6 minutes ?
7.
The reduction of a person’s oxygen consumption during periods of transcendental meditation may be looked upon as random
variable having the normal distribution with µ = 38.6 cc per minutes and σ = 6.5 cc per minute. Find the probabilities that during a
period of transcendental meditation a person’s oxygen consumption will be reduced by
a) at least 33.4 cc per minute;
b) at most 34.7 cc per minute.
8.
In a very large class in European history, the final examination grades have a mean of 71.6 and a standard deviation of 12.6. If it is
reasonable to approximate the distribution of these grades with a normal distribution, what percentage of the grades should exceed
79 ?
9.
If the amount of time a tourist spends in a famous museum is a random variable having the normal distribution with µ = 43.4
minutes and σ = 6.8 minutes, find the probabilities that a tourist will spend
a) at most 36.0 minutes in the museum;
b) anywhere from 40.0 to 50.0 minutes in the museum.
10.
According to Statistics Canada, Canadians aged 26 to 44 spend an average of 22 minutes traveling to work. Assume that such
travel times for workers aged 26 to 44 are normally distributed with a mean of 22 minutes and a standard deviation of 5 minutes.
a)
Find the probability that a randomly selected 26-to-44 year old Canadian worker spent at least 30 minutes traveling to work?
b) What percentage of such workers spent between 10 and 18 minutes traveling to work?
11.
Reaction time is normally distributed with a mean of 0.7 seconds and a standard deviation of 0.1 seconds. Find the probability that
an individual selected at random has a reaction time
a)
Greater than 0.9 seconds.
b)
Less than 0.6 seconds.
c)
Between 0.6 and 0.9 seconds.
12.
The IQ of students at St. Lawrence College were measured recently and found to be normally distributed with a mean of 100 and a
standard deviation of 15. What is the probability that a student selected at random will have an IQ
a)
Of 140 or higher?
b) Of 120 or higher?
c)
Between 100 and 120?
d) Of 90 or less?
13.
The average annual salary of a worker in a certain industry is $26 362. If we assume that the annual salaries are normally
distributed with a standard deviation of $6500, find the following
a)
What percentage earn below $15000?
b) What percentage earn above $40000?
14.
A normal distribution has the mean µ = 62.4. Find its standard deviation if 20.05 percent of the area under the curve lies to the
right of 79.2.
15.
The weights of ripe watermelons grown at Mr. Smith’s farm are normally distributed with a standard deviation of 2.8 lb. Find the
mean weight of Mr. Smith’s ripe watermelons if only 3% weight less than 15 lb.
16.
A random variable has the normal distribution with σ = 10. If the probability that the random variable will take on a value less
than 82.5 is 82.12%, what is the probability that it will take on a value greater than 58.3 ?
17.
A random variable has a normal distribution with σ = 4.0. If the probability is 0.9713 that this random variable will take on a
value less than 77.6, what is the probability that it will take on a value between 65.0 and 68.0 ?
18.
The average time required to perform job A is 78.5 minutes with a standard deviation of 16.2 minutes, and the average time
required to perform job B is 103.2 minutes with a standard deviation of 11.3 minutes with a standard deviation of 11.3 minutes.
Assuming normal distributions, what proportion of the time will job A take longer than the average job B, and what proportion of
time will job B take less time than the average job A ?
Best of Luck !
Steve
ANSWERS:
1.
a)
b)
c)
0.4599
0.1210
0.2223
2.
a)
b)
c)
d)
0.3336
0.9564
0.9564
0.000
3.
a)
b)
c)
d)
93.32%
77.34%
29.57%
91.98%
4
a)
b)
c)
d)
3.51%
29.12%
26.45%
31.82%
5.
a)
b)
c)
d)
0.9251
0.7734
0.0934
0.6514
6.
a)
b)
0.2420
0.3472
7.
a)
b)
0.7881
0.2743
8.
0.2776
9.
a)
b)
0.1379
0.5255
10.
a)
b)
0.0548
20.37%
11.
a)
b)
c)
12.
a)
b)
c)
d)
0.0038
0.0918
0.4082
0.2514
13.
a)
b)
4.0%
1.8%
14.
σ = 20
15.
20.26 lb
16.
93.32% (µ = 73.3)
17.
µ = 70 ? 0.2029
18.
0.0643 and 0.0143 (0.4933)
0.0227
0.1587
0.8186